Compensating for hysteresis

ABSTRACT

A method and apparatus for compensating for hysteresis in a system, the method comprising: determining a required input to the system from an output of the system using the Preisach model with the input of the Preisach model corresponding to the output of the system, and with the output of the Preisach model corresponding to the input of the system. The system may be an adaptive optics system. The input x may be an input voltage of an actuator that deforms a mirror, and the output y may be a value of a displacement of a mirror.

CROSS REFERENCE TO RELATED APPLICATIONS

This is the U.S. National Phase of PCT/GB2010/051487, filed Sep. 7,2010, which claims priority to British Patent Application No. 0916245.4,filed Sep. 16, 2009 and European Patent Application No. 09275081.9,filed Sep. 16, 2009, each of which are incorporated by reference hereinin their entireties.

FIELD OF THE INVENTION

The present invention relates to compensating for hysteresis, and otherprocesses related to hysteresis.

BACKGROUND

The exhibition of hysteresis by certain materials used in certainsystems, for example those used in a deformable mirror, is known.

A known problem relating to hysteresis in deformable mirrors is asfollows. An input voltage is applied to an actuator in a deformablemirror, which causes it to change shape. When the input voltage isturned off, the actuator will return to its original shape in time, butin a slightly different manner. It is likely that the actuator willexperience a second applied input voltage before it has returned to itsoriginal position. This causes the actuator to expand and contract in anunintended fashion. Thus, there is a degree of inaccuracy in thedeformable mirror system.

Known methods of reducing the degree of this inaccuracy (i.e. methods ofcompensating for hysteresis in, for example, a deformable mirror)implement the Preisach model to model hysteresis in the material, andthen implement an Inverse Preisach model to reduce the inaccuracy causedby hysteresis. Conventional applications of the Inverse Preisach modelrequire large amounts of processing, for example much greater amounts ofprocessing than is typically required for the forward Preisach model.

Conventionally, the Inverse Preisach model is implemented using a linearinterpolation based inversion algorithm, which requires large amounts ofprocessing. Also, an increasing input to the system tends not toconsistently lead to an increasing output from the system. This cancause interpolation problems.

The remainder of this section introduces Preisach model terminology usedlater below in the description of embodiments of the present invention.

The Preisach model describes hysteresis in terms of an infinite set ofelementary two-valued hysteresis operators (hysterons).

FIG. 1 is a schematic graph showing a typical input-output loop of asingle two-valued relay hysteron, referred to hereinafter as a“hysteron”. The x-axis of FIG. 1 represents an input voltage to thesystem, and is hereinafter referred to as the “input x”. For example,the input x is the input voltage applied to an actuator that deforms adeformable mirror. The y-axis of FIG. 1 represents an output voltagefrom the system, and is hereinafter referred to as the “output y”. Forexample, the output y is the displacement by which a deformable mirroris deformed by an actuator that has received an input voltage, e.g. theinput x. FIG. 1 shows that the input x ranges from minus two to two.Also, the output y takes a value of zero or one. The ranges for theinput x and the output y are merely exemplary, and may be differentappropriate values. The output level of one corresponds to the systembeing switched ‘on’, and the output level of zero corresponds to thesystem being switched ‘off’. The zero output level is shown in FIG. 1 asa bold line and is indicated by the reference numeral 10. The outputlevel of one is shown in FIG. 1 as a bold line and is indicated by thereference numeral 12. FIG. 1 shows an ascending threshold α at aposition corresponding to x=1, and a descending threshold β at aposition corresponding to x=−1.

Graphically, FIG. 1 shows that if x is less than the descendingthreshold β, i.e. −2<x<−1, the output y is equal to zero (off). As theinput x is increased from its lowest value (minus two), the output yremains at zero (off) until the input x reaches the ascending thresholdα, i.e. as x increases, y remains at 0 (off) if −2<x<1. At the ascendingthreshold α, the output y switches from zero (off) to one (on). Furtherincreasing the input x from one to two has no change of the output valuey, i.e. the hysteron remains switched ‘on’. As the input x is decreasedfrom its highest value (two), the output y remains at one (on) until theinput x reaches the descending threshold β, i.e. as x decreases, yremains at 1 if −1<x<2. At the descending threshold β, the output yswitches from one (on) to zero (off). Further decreasing the input xfrom minus one to minus two has no change of the output value y, i.e.the hysteron remains switched ‘off’.

Thus, the hysteron takes the path of a loop, and its subsequent statedepends on its previous state. Consequently, the current value of theoutput y of the complete hysteresis loop depends upon the history of theinput x.

Within a material, individual hysterons may have varied α and β values.The output y of the system at any instant will be equal to the sum overthe outputs of all of the hysterons. The output of a hysteron withparameters α and β is denoted as ξ_(αβ)(x). Thus, the output y of thesystem is equal to the integral of the outputs over all possiblehysteron pairs, i.e.

$y = {\underset{\alpha \geq \beta}{\int\int}{\mu\left( {\alpha,\beta} \right)}{\xi_{\alpha\;\beta}(x)}{\mathbb{d}\alpha}{\mathbb{d}\beta}}$

where μ(α,β) is a weighting, or density, function, known as the Preisachfunction.

This formula represents the Preisach model of hysteresis. The input tothe system corresponds to the input of the Preisach model (these inputscorrespond to x in the above equation). The output of the systemcorresponds to the output of the Preisach model (these outputscorrespond to y in the above equation).

FIG. 2 is a schematic graph showing all possible α-β pairs for thehysterons in a particular material. All α and β pairs lie in a triangle20 shown in FIG. 2. The triangle 20 is bounded by: the minimum of theinput x (minus two); the maximum of the input x (two); and the line α=βline (since α≧β).

Increasing the input x from its lowest amount (minus 2) to a value x=u₁provides that all of the hysterons with an α value less than the inputvalue of u₁ will be switched ‘on’. Thus, the triangle 20 of FIG. 2 isseparated into two regions. The first region contains all hysterons thatare switched ‘on’, i.e. the output y equals a value of one. The secondregion contains all hysterons that are switched ‘off’, i.e. the output yequals a value of zero. FIG. 3 is a schematic graph showing the regionof all possible α-β pairs, i.e. the triangle 20, divided into the abovedescribed two regions by increasing the input x from its lowest amountto a value x=u₁. The first region, i.e. the region that contains allhysterons that are switched ‘on’ is hereinafter referred to as the“on-region 22”. The second region, i.e. the region that contains allhysterons that are switched ‘off’ is hereinafter referred to as the“off-region 24”.

Decreasing the input x from the value x=u₁ to a value x=u₂ provides thatall of the hysterons with a β value greater than the input value of u₂will be switched ‘off’. Thus, the on-region 22 and the off-region 24 ofthe triangle 20 change as the input x is decreased from the value x=u₁to the value x=u₂. FIG. 4 is a schematic graph showing the regions 22,24 of the triangle 20 formed by decreasing the input x from the valuex=u₁ to the value x=u₂, after having previously increased the input x asdescribed above with reference to FIG. 3.

An increasing input can be thought of as a horizontal link that movesupwards on the graph shown in FIGS. 2-4. Similarly, a decreasing inputcan be thought of as a vertical link that moves towards the left on thegraph shown in FIGS. 2-4.

By alternately increasing and decreasing the input x, the triangle 20 isseparated in to two regions, the boundary between which has a number ofvertices. FIG. 5 is a schematic graph showing the regions 22, 24 of thetriangle 20 formed by alternately increasing and decreasing the input x.Alternately increasing and decreasing the input x produces a “staircase”shaped boundary between the on-region 22 and the off-region 24,hereinafter referred to as the “boundary 300”. The boundary 300 has fourvertices, referred to hereinafter as the “first x-vertex 30” (which hascoordinates (α₁,β₁), the “second x-vertex 32” (which has coordinates(α₂,β₁)), the “third x-vertex 34” (which has coordinates (α₂,β₂)), andthe “fourth x-vertex 36” (which has coordinates (α₃,β₂)).

In FIG. 5 the final voltage change in the input x is a decreasingvoltage change (as indicated by the vertical line from the thirdx-vertex 34 to the line α=β). However, it is possible for the finalvoltage change in the input x to be an increasing voltage change. Thiscould be considered to be followed by a decreasing voltage change ofzero for convenience.

For the Preisach Model to represent a material's behaviours, thematerial has to have the following two properties: the material musthave the wiping-out property, which provides that certain increases anddecreases in the input x can remove or ‘wipe-out’ x-vertices; and thematerial must have the congruency property, which states that all minorhysteresis loops that are formed by the back-and-forth variation ofinputs between the same two extremum values are congruent.

The output y of the system is dependent upon the size and shape of theon-region 22. The on-region 22, in turn, is dependent upon thex-vertices 30, 32, 34, 36. Thus, as described in more detail laterbelow, the output y of the system can be determined using the x-vertices30, 32, 34, 36 of the boundary between the on-region 22 and theoff-region 24.

The output y of the system illustrated by FIG. 5 is:

$y = {{\underset{\alpha \geq \beta}{\int\int}{\mu\left( {\alpha,\beta} \right)}{\xi_{\alpha\;\beta}(x)}{\mathbb{d}\alpha}{\mathbb{d}\beta}} = {{\underset{{on}\text{-}{region}}{\int\int}{\mu\left( {\alpha,\beta} \right)}{\xi_{\alpha\;\beta}(x)}{\mathbb{d}\alpha}{\mathbb{d}\beta}} + {\underset{{off}\text{-}{region}}{\int\int}{\mu\left( {\alpha,\beta} \right)}{\xi_{\alpha\;\beta}(x)}{\mathbb{d}\alpha}{\mathbb{d}\beta}}}}$In the on-region 22, all hysterons are switched on, and thereforeξ_(αβ)(x)=1. Similarly, in the off-region 24 all hysterons are switchedoff, and therefore ξ_(αβ)(x)=0. Thus,

$y = {{{\underset{{on}\text{-}{region}}{\int\int}{{\mu\left( {\alpha,\beta} \right)} \cdot 1 \cdot {\mathbb{d}\alpha}}{\mathbb{d}\beta}} + {\underset{{off}\text{-}{region}}{\int\int}{{\mu\left( {\alpha,\beta} \right)} \cdot 0 \cdot {\mathbb{d}\alpha}}{\mathbb{d}\beta}}} = {\underset{{on}\text{-}{region}}{\int\int}{\mu\left( {\alpha\;,\beta} \right)}{\mathbb{d}\alpha}{\mathbb{d}\beta}}}$

By considering the x-vertices on the boundary 300, it can be shown thatthe integral can be estimated as follows:

$y = {\sum\limits_{k = 1}^{n}\left( {y_{\alpha_{k}\beta_{k}} - y_{\alpha_{k}\beta_{k - 1}}} \right)}$where:

-   -   y_(α) _(i) _(β) _(j) is the output y resulting from increasing        the input voltage x from the minimum to α_(i) and then        decreasing it to β_(j);    -   β₀ is the minimum saturation voltage, i.e. minus two; and    -   n is the number of vertical trapezia formed by the x-vertices on        the boundary 300, i.e. n is therefore equal to └½×number of        vertices┘.

In practice, to calculate the above equation, values of y_(αβ) for anumber of points in the triangle 20 are generated. Typically, a value ofy_(αβ) for each α-β pairs in a grid of α-β pairs in the triangle 20 iscalculated. This is done by increasing the input x from its minimum(minus two) to α, and then decreasing it to β, and measuring the outputy of the system. For α-β pairs not on the grid, a value of y_(αβ) isfound using bilinear interpolation, or linear interpolation, using α-βpairs on the grid.

SUMMARY OF THE INVENTION

In a first aspect the present invention provides a method ofcompensating for hysteresis in a system, the method comprising:determining a required input to the system from an output of the systemusing the Preisach model with the input of the Preisach modelcorresponding to the output of the system, and with the output of thePreisach model corresponding to the input of the system.

Using the Preisach model with the input of the Preisach modelcorresponding to the output of the system, and with the output of thePreisach model corresponding to the input of the system, may comprisedetermining a value of an input x to the system using the followingformula:

$x = {\underset{\gamma \geq \delta}{\int\int}{\lambda\left( {\gamma,\delta} \right)}{ɛ_{\gamma\;\delta}(y)}{\mathbb{d}\gamma}{\mathbb{d}\delta}}$where: y is an output of the system; γ is the level to which the outputy is increased as the input x is increased; δ is the level to which theoutput y is decreased as the input x is decreased; λ(γ,δ) is a densityfunction; and ε_(γδ)(y) is the output of an imaginary hysteron havingparameters γ and δ.

Using the Preisach model with the input of the Preisach modelcorresponding to the output of the system, and with the output of thePreisach model corresponding to the input of the system, may comprisedetermining a value of an input x to the system using the followingformula:

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$where: x_(γ) _(i) _(δ) _(j) is the input of the system resulting fromincreasing the output y from the minimum output to γ_(i) and thendecreasing it to δ_(j); and δ₀ is the minimum output.

The formula

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k\;}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$may be calculated by: determining a value for x_(γδ) for each γ-δ pairin a discrete set of γ-δ pairs; and for all required values of x_(γ)_(i) _(δ) _(j) where γ_(i) and δ_(j) are not in the discrete set,determining the value of x_(γ) _(i) _(δ) _(j) using a process ofinterpolation using values of x_(γδ) where γ and δ are in the discreteset.

The system may be an adaptive optics system.

The input x may be an input voltage of an actuator that deforms adeformable mirror, and the output y may be a value of a displacement ofa deformable mirror.

The method may further comprise a process of determining an updatedrequired input to the system, the determining process comprising:measuring a value of the input to the system; and determining theupdated required input to the system using the determined required inputto the system and the measured required input to the system.

The method may further comprise a process of determining an updatedrequired input to the system, the determining process comprising:measuring a value of the output of the system to determine a measuredoutput value corresponding to the determined required input; anddetermining the updated required input to the system using the measuredoutput value and the output of the system.

In a further aspect the present invention provides an apparatus adaptedto perform the method of any of any of the above aspects.

In a further aspect the present invention provides a computer program orplurality of computer programs arranged such that when executed by acomputer system it/they cause the computer system to operate inaccordance with the method of any of the above aspects.

In a further aspect the present invention provides a machine readablestorage medium storing a computer program or at least one of theplurality of computer programs according to the above aspect.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic graph showing a typical input-output loop of asingle two-valued relay hysteron;

FIG. 2 is a schematic graph showing all possible α-β pairs for thehysterons in a particular material;

FIG. 3 is a schematic graph showing the region of all possible α-βpairs;

FIG. 4 is a schematic graph showing the on-region and off-region of theregion of all possible α-β pairs formed by decreasing the input x fromthe value x=u₁ to the value x=u₂;

FIG. 5 is a schematic graph showing the on-region and off-region of theregion of all possible α-β pairs formed by alternately increasing anddecreasing the input x;

FIG. 6 is a schematic illustration of a mirror interrogation system;

FIG. 7 is a schematic graph showing the space of possible increases anddecreases in the output y generated by alternately increasing anddecreasing the input x;

FIG. 8 is a process flow chart showing a method of implementing anInverse Preisach model according to an embodiment of the presentinvention;

FIG. 9 is a schematic graph showing a grid of γ-δ pairs defined on thespace of all possible γ-δ pairs; and

FIG. 10 is a schematic illustration of an adaptive optics system.

DETAILED DESCRIPTION

FIG. 6 is a schematic illustration of a mirror interrogation system 99in which an Inverse Preisach model is performed. The Inverse Preisachmodel is implemented in an embodiment of compensating for hysteresis, asdescribed later below with reference to FIG. 10, after the descriptionof the Inverse Preisach model.

The mirror interrogation system 99 comprises a controller 100, adeformable mirror 101, a beam-splitter 106, and a wave-front sensor 108.The deformable mirror 101 comprises an actuator 102 and a mirror 104.

The controller 100 comprises an output and an input. The output of thecontroller 100 is connected to the actuator 102. The input of thecontroller is connected to the wave-front sensor 108. The controller 100receives a signal from the wave-front sensor 108. The controllerprocesses the signal received from the wave-front sensor 108, asdescribed in more detail later below. The controller 100 sends a controlsignal to the actuator 102. The control signal depends on the signalreceived by the controller 100 from the wave-front sensor 108, asdescribed in more detail later below.

The actuator 102 comprises an output and an input. The input of theactuator 102 is connected to the controller 100. The output of theactuator 102 is connected to the mirror 104. The actuator 102 receivesthe control signal from the controller 100. The actuator changes theshape of, or deforms, the mirror 104 via the actuator output, dependingon the received control signal. In this embodiment, the control signalreceived by the actuator is an input voltage. This input voltage to theactuator corresponds to the input x for a Preisach model, as describedabove and referred to later below. The value of the input x determinesthe amount by which the actuator 102 deforms the mirror 104.

The mirror 104 is deformed by the actuator 102. The displacement of themirror, i.e. the amount by which the mirror 104 is deformed by theactuator 102, corresponds to the output y for a Preisach model, asdescribed above and referred to later below.

In this embodiment, a beam of light is reflected and split by the mirrorinterrogation system 99. The beam of light is indicated by arrows inFIG. 6. For clarity and ease of understanding, the beam of light isshown as separate sections: a first section of the beam of light that isincident on the mirror 104, (the first section is hereinafter referredto as the “incident beam 110”); a second section of the beam of lightthat is reflected from the mirror 104 and is incident on thebeam-splitter 106 (the second is hereinafter referred to as the“reflected beam 112”); and a third section and a fourth section of thebeam of light that are formed by the beam splitter 106 splitting thereflected beam (the third section is hereinafter referred to as the“image beam 114” and the fourth section is hereinafter referred to asthe “feedback beam 116”).

The beam splitter splits the reflected beam 106 into the image beam 114and the feedback beam 116.

The feedback beam 116 is incident on the wave-front sensor 108.

The wave-front sensor 108 detects the feedback beam 116. The wave-frontsensor 108 measures a value of the curvature of the mirror 104. Thewave-front sensor 108 comprises an output. The wave-front sensor 108sends a signal corresponding to the detected feedback beam 116, i.e. asignal corresponding to the curvature of the mirror 104, to thecontroller 100 via the output of the wave-front sensor 108.

The controller 100 receives the signal corresponding to the detectedfeedback beam 116 from the wave-front sensor 108. In this embodiment,the controller 100 comprises a processor (not shown). The processor usesthe signal corresponding to the detected feedback beam 116 to determinea value corresponding to the displacement of the mirror 104, i.e. avalue for the output y of the Preisach model. The processor furthergenerates a control signal, i.e. the input x for the Preisach model,using the determined output y. The control signal is sent to theactuator 102, and the actuator 102 deforms the mirror 104 depending onthe received control signal as described above.

An embodiment in which an Inverse Preisach model is performed, in themirror interrogation system 99 described above with reference to FIG. 6,will now be described. In this embodiment, a problem of obtaining aninverse result for the Preisach model is addressed making use of aspectsof the forward Preisach model, as opposed to using the conventionalapproach of interpolation-based techniques.

In this embodiment, the deformable mirror 101 has the wiping-outproperty as described above for the forward Preisach model, anddescribed below for this embodiment.

As described in more detail above, alternately increasing and decreasingthe input x (which in this embodiment is the input voltage applied tothe actuator 102) produces the α-β graph shown in FIG. 5. The α-β graphshown in FIG. 5 shows the first x-vertex 30, the second x-vertex 32, thethird x-vertex 34, and the fourth x-vertex 36.

The output y (which in this embodiment is the displacement of the mirror104 by the actuator 102) of the system is, in general, an increasingfunction with respect to the input x, i.e. as the input x increases, theoutput y increases, and as the input x decreases, the output ydecreases. Thus, alternately increasing and decreasing the input x,produces alternating increases and decreases in the output y. This isshown schematically in FIG. 7.

FIG. 7 is a schematic graph showing the space of possible increases anddecreases in the output y generated by the increasing and decreasing ofthe input x. The vertical axis of FIG. 7 is the level to which theoutput y is increased as the input x is increased, indicated by thereference sign γ. The horizontal axis of FIG. 7 is the level to whichthe output y is decreased as the input x is decreased, indicated by thereference sign δ. The space of all possible γ-δ pairs is indicated inFIG. 7 by the reference number 21. The space of γ-δ pairs 21 correspondsto the triangle 20 of all possible α-β pairs for the hysterons in aparticular material, as described above with reference to FIGS. 2-5. Thespace of γ-δ pairs 21 is bounded by: the minimum of the output y (zero);the maximum of the output (one); and the line γ=δ line (since γ≧δ).

FIG. 7 further shows the output y that is produced by the input x beingalternately increased and decreased according to the α-β graph shown inFIG. 5. In this embodiment, the output y comprises four vertices,hereinafter referred to as the “first y-vertex 40” (which hascoordinates (γ₁, δ₁)), the “second y-vertex 42” (which has coordinates(γ₂, δ1)), the “third y-vertex 44” (which has coordinates (γ₂, δ₂)), andthe “fourth y-vertex 46” (which has coordinates (γ₃, δ₂)).

The first x-vertex 30 of the boundary 300 shown in the α-β graph for theinput x (FIG. 5) corresponds to the first y-vertex 40 for the output yshown in FIG. 7. In other words, as the input x is increased to α₁, theoutput y correspondingly increases to γ₁. Then, as the input x isdecreased to β₁, the output y correspondingly decreases to δ₁.

The second x-vertex 32 of the boundary 300 shown in the α-β graph forthe input x (FIG. 5) corresponds to the second y-vertex 42 for theoutput y shown in FIG. 7. In other words, as the input x is decreased toβ₁, the output y correspondingly decreases to δ₁. Then, as the input xis increased to α₂, the output y correspondingly increases to γ₂.

The third x-vertex 34 of the boundary 300 shown in the α-β graph for theinput x (FIG. 5) corresponds to the third y-vertex 44 for the output yshown in FIG. 7. In other words, as the input x is increased to α₂, theoutput y correspondingly increases to γ₂. Then, as the input x isdecreased to δ₂, the output y correspondingly decreases to δ₂.

The fourth x-vertex 36 of the boundary 300 shown in the α-β graph forthe input x (FIG. 5) corresponds to the fourth y-vertex 46 for theoutput y shown in FIG. 7. In other words, as the input x is decreased toδ₂, the output y correspondingly decreases to δ₂. Then, as the input xis increased to α₃, the output y correspondingly increases to γ₃.

In this embodiment, the wiping-out property, which holds for the α-βgraph for the input x shown in FIG. 5, also holds for the γ-δ graph forthe output y shown in FIG. 7. For example, if the input x is decreasedfrom its value at the fourth x-vertex 36 to the β-value value of thesecond x-vertex 32, i.e. to β₁, then the second, third and fourthx-vertices 32, 34, 36 are wiped out. Thus, the output y is onlydependent on the first x-vertex 30. The first x-vertex 30 corresponds tothe first y-vertex 40 in FIG. 7, and so the output y depends only on thefirst y-vertex 40, i.e. the second, third, and fourth y-vertices 42, 44,46 have been wiped out. A similar argument can be constructed for anincreasing value of the input x. In this embodiment, the wiping-outproperty holds if y is an increasing function with respect to x. Inpractice this requirement can be assumed to hold (even though nomaterial is perfect and therefore y is not quite an increasing function)because the material properties are such that useful results arenevertheless produced.

In this embodiment, the congruency property, which holds for the α-βgraph for the input x shown in FIG. 5, does not necessarily hold truefor the inverse. For example, the output y is to be increased from y₁ toy₂. To do this, the input x is increased from x₁ to x₂. To return theoutput y to y₁, the input x is decreased to x₁. Thus, each time theoutput y is increased from y₁ to y₂ and decreased to y₁ again, anidentical ‘loop’ of input values values is created. However, the input x(and/or the output y) may have a history that requires a different valueof x₁ to bring the output to y₁, and a different value of x₂ to bringthe output to y₂. x₂ and x₁ are not necessarily the same, thusincreasing the output y from y₁ to y₂ and back to y₁ may not result incongruent ‘loops’ of the input x values for all input histories. Inother words, the range over which the input x is varied to produce aparticular loop in the output y changes with different input histories.

In this embodiment, it is not necessary for the congruency property tohold for the inverse because, in practice, the material tends not tohave a perfect congruency property for the forward Preisach model. Also,minor hysteresis loops near the centre of the major hysteresis loop arelikely to all be very similar. In practice, the better the congruencyproperty holds, the better the inverse Preisach model, herein described,will work. In other words, the material does not have a perfect‘forward’ congruency property, so the ‘inverse’ congruency propertytends not be detrimentally limited. Also, for readings in the middle ofthe hysteresis loop, all of the loops are quite similar. Thus, thecongruency property tends to hold reasonably well in this middle range.

The formula for the inverse model is:

$x = {\underset{\gamma \geq \delta}{\int\int}{\lambda\left( {\gamma,\delta} \right)}{ɛ_{\gamma\delta}(y)}{\mathbb{d}\gamma}{\mathbb{d}\delta}}$Where:

-   -   λ(γ,δ) is a density function; and    -   ε_(γδ)(y) is the output of an imaginary hysteron having        parameters γ and δ.

This formula represents an Inverse Preisach model of hysteresis for thepresent embodiment. The input to the system, i.e. the input voltage tothe actuator 102, corresponds to the output of the Inverse Preisachmodel (these correspond to x in the above equation). The output of thesystem, i.e. the displacement of the mirror 104, corresponds to theinput of the Inverse Preisach model (these correspond to y in the aboveequation).

This formula can be rewritten as a summation, in the same way as for theforward model:

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$where:

-   -   x_(γ) _(i) _(δ) _(j) is the input x resulting from increasing        the output y from the minimum (in this embodiment, the minimum        output is zero) to γ_(i) and then decreasing it to δ_(j); and    -   δ₀ is the minimum output, i.e. zero.    -   n is the number of vertical trapezia formed by the y-vertices in        triangle 21, and is therefore equal to [½×number of vertices].

As described above for the forward Preisach model, values of x_(γδ)where γ and/or δ are not on the grid, is found using bilinear or linearinterpolation using determined values of x_(γδ) where γ and δ are on thegrid.

In this embodiment, the minimum output is 0. However, in otherembodiments, the minimum output is not zero. In embodiments in which theminimum output is not zero, an appropriate additional constant is addedto the sum/integral in the above equations for the inverse Preisachmodel to account for the non-zero minimum output.

In a corresponding way to performing the forward Preisach model, a valueof the input x is generated for each γ-δ pair in a grid of γ-δ. Thesevalues are hereinafter denoted as x_(γδ). For γ-δ pairs not on the grid,a value of x is found using bilinear interpolation, or linearinterpolation, using γ-δ pairs on the grid.

In this embodiment, the grid input values are generated using theforward Preisach model. The input x is increased slowly (i.e. in smallincrements) until y=γ, and then the input x is slowly decreased untily=β. This embodiment of implementing an Inverse Preisach model isdescribed in more detail later below, with reference to FIG. 7. Theforward Preisach model, utilised as described above, advantageouslytends not to suffer from creep. Also, the forward Preisach modeladvantageously tends to be time-independent, and have perfect wiping-outand congruency properties. In other embodiments, the grid of inputvalues x_(γδ) could be generated by implementing the above method usingthe deformable mirror instead of the forward Preisach Model. However,the deformable mirror may suffer from creep and may not perfectlysatisfy the Preisach criteria, i.e. the deformable mirror may not haveperfect wiping-out and congruency properties.

FIG. 8 is a process flow chart showing a method of implementing anInverse Preisach model according to the above described embodiment.

At step s2, a grid of γ-δ pairs is defined on the space of all possibleγ-δ pairs. In other words, a grid of points is defined on the space ofγ-δ pairs 21 shown in FIG. 7.

FIG. 9 is a schematic graph showing a grid of γ-δ pairs (indicated bydots in the space of γ-δ pairs 21) defined on the space of all possibleγ-δ pairs (the space of γ-δ pairs 21).

At step s4, for a particular γ-δ pair, the input x to the system isslowly increased from the minimum input (minus two) until the output yof the system equals the value of γ of the particular γ-δ pair.

At step s6, for the particular γ-δ pair, the input x to the system isslowly decreased until the output y of the system equals the value of δof the particular γ-δ pair.

At step s8, the value of the input x after having performed the steps s4and s6 above is stored in a table for that particular γ-δ pair, i.e. thevalue of the input x is stored as x_(γδ), as described above.

At step s10, the steps s4, s6 and s8 are repeated for all remaining γ-δpairs. Thus, for each value of the output y that is defined as a pointon the grid of γ-δ pairs in FIG. 8, a corresponding value x_(γδ) of theinput x that is required to produce such an output γ, is determined andstored.

In this embodiment, steps s2-s10, as described above, are performed onceand the grid of input values x_(γδ) is stored and used as a referencefor performing step s12.

At step s12, the input x for a series system of outputs is calculatedusing the formula (described earlier above):

$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$where:

-   -   x_(γ) _(i) _(δ) _(j) the input x resulting from increasing the        output y from the minimum to γ_(i) and then decreasing it to        δ_(j) (zero);    -   δ₀ is the minimum output, i.e. zero; and    -   n is the number of vertical trapezia formed by the y-vertices in        triangle 21, and is therefore equal to └½×number of vertices┘.

A value of x_(γδ) for values of γ and/or δ not on the grid, is foundusing bilinear interpolation, or linear interpolation, using determinedgrid values of x_(γδ).

Thus, a method of performing an Inverse Preisach model is provided. Theprovided inverse procedure tends to be faster than conventional,iterative inverse procedures.

A further advantage is that algorithms prepared for the forward Preisachmodel tend to be usable (with the alternative grid of points) forimplementing the provided inverse procedure.

A further advantage of above described Inverse Preisach model is thatthe grid of γ-δ pairs provided by the process can be updated usingfeedback from sensors, for example the wave-front sensor 108. This tendsto allow for faster processing and more accurate estimations.

The above described Inverse Preisach model is used to compensate forhysteresis in a system. In the following embodiment, the InversePreisach model is used to compensate for hysteresis in an adaptiveoptics system.

The adaptive optics system described in the following embodimentcomprises the same deformable mirror 101 (i.e. the same actuator 102 andmirror 104) present in the mirror interrogation system 99. This is sothat the inverse Preisach information generated by the mirrorinterrogation system 99 (as described above) can be used to compensatefor hysteresis in the adaptive optics system, i.e. the inverse Preisachinformation is derived from the particular deformable mirror 101.Alternatively, in other embodiments a deformable mirror of the same typeas the deformable mirror that has been interrogated and had inversePreisach information generated for it, for example a deformable mirrormanufactured to the same specifications as those of the interrogateddeformable mirror, is used. In other embodiments, a deformable mirror isarranged as part of a mirror interrogation system and as part of anadaptive optics system at the same time.

FIG. 10 is a schematic illustration of an adaptive optics system 150 inwhich an embodiment of compensating for hysteresis is implemented.

The adaptive optics system 150 comprises the controller 100, thedeformable mirror 101, a further beam splitter 152, and a furtherwave-front sensor 154. The deformable mirror 101 comprises the actuator102 and the mirror 104.

In this embodiment, a beam of light is split and reflected by theadaptive optics system 150. The beam of light is indicated by arrows inFIG. 10. For clarity and ease of understanding, the beam of light isshown as separate sections: a first section of the beam of light that isincident on the further beam splitter 152, (the first section ishereinafter referred to as the “further incident beam 156”); a secondand a third section of the beam of light that are formed by the furtherbeam splitter 152 splitting the further incident beam 156. The secondsection is hereinafter referred to as the “mirror beam 157”, and isreflected by the mirror 104. The third section is hereinafter referredto as the “sensor beam 158”) and is incident on the further wave-frontsensor 154.

In this embodiment, the actuator 102 deforms the mirror 104 to generatea spherical surface to enable the system to correct for sphericalaberrations.

The further wave-front sensor 154 detects the sensor beam 158. Thefurther wave-front sensor 154 sends a signal corresponding to thedetected sensor beam 158, i.e. a signal corresponding to the furtherincident beam 156, to the controller 100.

The controller 100 receives the signal corresponding to the detectedsensor beam 158 from the further wave-front sensor 154. The processor inthe controller uses the signal corresponding to the detected sensor beam158 to determine a value corresponding to the displacement of the mirror104, i.e. a value for the output y of the Preisach model. The processorfurther generates a control signal using the above described InversePreisach model, i.e. the input x using the determined output y. Thecontrol signal is sent to the actuator 102, and the actuator 102 deformsthe mirror 104 depending on the received control signal. In this way,the hysteresis experienced by the mirror 104 resulting from operation inresponse to the further incident beam 156 is compensated for.

In the above embodiments, the light can be of any wavelength, forexample infra-red.

In the above described embodiment, a grid of γ-δ pairs is defined. Inthis embodiment, the spacing between the grid pairs is small. In theabove described embodiment 8001 grid points are implemented. Typically,the larger the number of grid points, the more accurate the estimates ofthe input x to the system are.

In the above embodiments, the output y (shown in FIG. 7) produced whenthe input x is alternately increased and decreased according to the α-βgraph shown in FIG. 5, comprises four vertices. However, in otherembodiments, the input x is alternately increased and decreased in adifferent manner, i.e. the α-β graph for input x has a different numberof vertices. Thus, in other embodiments, the output y produced by theinput x is different, for example, the output y may have a differentnumber of vertices in the γ-δ space.

In the above embodiments, the congruency property does not hold for theoutput y. However, in other embodiments, the congruency property doeshold for the output y.

In the above embodiments, the grid of γ-δ pairs, as shown in FIG. 9, isa rectangular grid. However, in other embodiments the points of the γ-δgrid, i.e. the γ-δ pairs, are distributed in a different appropriatemanner. For example, in other embodiments the grid of γ-δ pairs is atriangular grid. The process of bilinear or linear interpolation used todetermine values of x_(γδ) for values of γ and/or δ not on the grid, ismodified accordingly, or a different appropriate process is used, toaccount for the grid of γ-δ pairs.

In the above described embodiments, the values of x_(γδ) for each γ-δpair on the grid of γ-δ pairs are determined one at a time, as describedabove with reference to steps s4-s10 of the above described method, andFIG. 8. However, in other embodiments some or all of the values ofx_(γδ) for each γ-δ pair on the grid of γ-δ pairs are determinedconcurrently, for example by implementing steps s4-s8 of the abovedescribed method concurrently on different specimens of a particularmaterial.

In the above embodiments, the values of x_(γδ) for each γ-δ pair on thegrid of γ-δ pairs are determined by increasing the output to γ, thendecreasing the output to δ. However, in other embodiments, the values ofx_(γδ) are determined in a different manner. For example, a series ofx_(γδ) values could be found by increasing the output to a particularvalue, and then decreasing the output to a series of values, each valuelower than the last.

Returning to FIG. 6, the controller 100 implements inter alia thevarious method steps described above. The controller may be implementedor provided by configuring or adapting any suitable apparatus, forexample one or more computers or other processing apparatus orprocessors, and/or providing additional modules. The apparatus maycomprise a computer, a network of computers, or one or more processors,for implementing instructions and using data, including instructions anddata in the form of a computer program or plurality of computer programsstored in or on a machine readable storage medium such as computermemory, a computer disk, ROM, PROM etc., or any combination of these orother storage media.

A further optional process of adaptively updating the Inverse Preisachmodel may be incorporated into the above described Inverse Preisachprocess to provide a further embodiment of an Inverse Preisach process.This further embodiment will now be described. In this furtherembodiment, the deformable mirror is arranged as part of the mirrorinterrogation system and as part of the adaptive optics system at thesame time.

The above described formula used to determine the input x for a seriessystem of outputs is adapted as follows:

$\begin{matrix}{x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}} \\{= {\left\lbrack {{\sum\limits_{k = 1}^{n - 1}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)} - x_{\gamma_{n}\delta_{n - 1}}} \right\rbrack + x_{\gamma_{n}\delta_{n}}}}\end{matrix}$where x is the estimation of the input to the system based on the abovedescribed embodiment of an Inverse Preisach model. Thus, the followingformula holds:

$x_{A} = {\left\lbrack {{\sum\limits_{k = 1}^{n - 1}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)} - x_{\gamma_{n}\delta_{n - 1}}} \right\rbrack + \left\lbrack {x_{\gamma_{n}\delta_{n}} + \left( {x_{A} - x} \right)} \right\rbrack}$where x_(A) is the actual input to the system (required to produce thedesired output).

Thus, adjusting the value of x_(γ) _(n) _(δ) _(n) by an amountproportional to (x_(A)−x) increases the accuracy of the estimation ofthe input to the system x based on the above described embodiment of anInverse Preisach model. If the γ-δ pair (γ_(n), δ_(n)) is a grid point,then x_(γ) _(n) _(δ) _(n) is adjusted by an amount proportional to(x_(A)−x). However, in this embodiment, the γ-δ pair (γ_(n), δ₂) isunlikely to be a grid point on the above described grid of γ-δ pairs.Therefore, the grid points surrounding the γ-δ pair (γ_(n), δ_(n)) areeach adjusted by an amount equal to k(x_(A)−x), where k is a constant.In this embodiment k=0.005. The value k=0.005 tends to provide the mostimproved estimate of the input x for the above described embodiment ofthe Inverse Preisach model. The value of x_(γ) _(n) _(δ) _(n) iscalculated as described above using bilinear or linear interpolation.

In practice, it may not be possible to determine a value of x_(A).However, the output y as a function of the input x tends to be smooth.Thus, the value of k(x_(A)−x) may be estimated by the following term:K(y_(A)−δ_(n))where:

-   -   K is a constant; and    -   y_(A) is the actual output of the system (after applying voltage        x); and    -   δ_(n) is the required output.

The actual output of the system y_(A) may be determined by anyappropriate means. For example, the actual output of the system y_(A)may be determined by directly coupling a strain gauge or a capacitivesensor to the mirror 104, or by implementing the mirror interrogationsystem 99, as described above with reference to FIG. 6, i.e. by using awave-front sensor.

In the above embodiments, the grid points surrounding the γ-δ pair(y_(n), δ_(n)) are each adjusted by an amount equal to k(x_(A)−x) (orK(y_(A)−δ_(n))). For example, the four points surrounding the particulargrid point (γ_(n), δ_(n)) are each adjusted. In other examples adifferent number of points can be adjusted. For example, in otherembodiments more than four points surrounding the particular grid point(γ_(n), δ_(n)) are each adjusted by different amounts. An advantageprovided by this is that more accurate estimations tend to be produced.

The above described adaptive updating process tends to provide that agrid of γ-δ pairs in which the spacing between the grid pairs is small,is not necessary. Indeed, the adaptive updating process tends to providemore accurate results using a grid of γ-δ pairs in which the spacingbetween the grid pairs is larger. In this embodiment, 351 γ-δ pairs areutilised. This tends to advantageously allow for faster computation ofthe estimated values.

In the above embodiments, the value of the constant k used in theadaptive updating process is 0.005. However, in other embodimentsdifferent values of k are used.

The above described adaptive updating process advantageously providesthat the grid of γ-δ pairs (or the grid of α-β pairs) used in theprocess can be updated using feedback from sensors. This tends toprovide more accurate estimations.

In the above embodiments, the controller comprises a processor whichuses the signal corresponding to the detected feedback beam to determinea value corresponding to the displacement of the deformable mirror, i.e.a value for the output y of the Preisach model. However, in otherembodiments the value corresponding to the displacement of thedeformable mirror is determined in a different appropriate way. Forexample, in other embodiments the processor determines the displacementof the deformable mirror using signals from sensors that directlymeasure the displacement of the deformable mirror.

In the above embodiments, the deformable mirror comprises a mirror thatis deformed by an actuator. However, in other embodiments the deformablemirror is a different appropriate type of deformable mirror, for examplea bimorph mirror.

In a further embodiment, the actuator 102 is used to control theposition of the mirror 104 to generate a piston action, correctingphase. In a further embodiment, a discrete array of such phasecorrectors is used to generate a multi-element deformable mirror. Eachdiscrete corrector can be controlled as described above.

In the above embodiments, the output y of the Preisach model is a valueof the displacement of the deformable mirror. However, in otherembodiments the output y of the Preisach model is a differentappropriate parameter. For example, in other embodiments the output isthe measured value of the feedback beam detected by the wave-frontsensor.

In the above embodiments, the input x of the Preisach model is a valueof the input voltage (control signal) received by the actuator. However,in other embodiments the input x of the Preisach model is a differentappropriate parameter.

In the above embodiments, the processor generates the control signal,i.e. the input x for the Preisach model, using the determined output y.However, in other embodiments the control signal is generated usingdifferent means, or a combination of means. For example, in otherembodiments the control is determined from a user input.

In the above embodiments, a wave-front sensor is used to provide thesignal corresponding to the detected feedback beam, i.e. a signalcorresponding to the curvature of the mirror. However, in otherembodiments a different appropriate device is used. For example, astrain gauge or capacitive sensor directly coupled to the mirror couldbe used.

In the above embodiments, hysteresis is compensated for in a deformablemirror of an adaptive optics system. However, in other embodiments,hysteresis is compensated for in any appropriate material of theadaptive optics system. Also, in other embodiments, hysteresis iscompensated for in other materials of other appropriate systems, forexample systems other than optics systems. In these embodiments, theinput and output of the Preisach model are different appropriateparameters.

The invention claimed is:
 1. A method of compensating for hysteresis ina system, the method comprising: providing a system, the systemcomprising an actuator and one or more components that exhibithysteresis including a deformable material, the components that exhibithysteresis having a wiping-out property, the system comprising a systeminput and a system output, the system output being dependent on thesystem input; determining an inverse Preisach model for the system, theinverse Preisach model comprising a model input and a model output, themodel output being dependent on the model input, the model inputcorresponding to the system output, the model output corresponding tothe system input; ascertaining a desired system output; determiningusing the desired system output and the inverse Preisach model for thesystem, a required system input corresponding to the desired systemoutput; and applying, to the system, the determined required systeminput to deform the deformable material using the actuator; wherein thestep of determining the inverse Preisach model for the system comprises:defining a plurality of pairs of values, each pair of values comprisinga first value and a second value, each of the first and second valuesbeing less than or equal to a maximum system output and greater than orequal to a minimum system output, the first value being greater than orequal to the second value within each pair; and for each pair of values,determining a system input corresponding to that pair of values, by:increasing the system input, from the minimum system input, until thesystem output is equal to the first value corresponding to therespective pair of values; and thereafter decreasing the system inputuntil the system output is equal to the second value corresponding tothe respective pair of values, the value that the system input isdecreased to being the determined system input corresponding to thatpair of values.
 2. A method according to claim 1, wherein determiningthe required system input comprises determining a value of an input x tothe system using the following formula:$x = {\underset{\gamma \geq \delta}{\int\int}{\lambda\left( {\gamma,\delta} \right)}{ɛ_{\gamma\;\delta}(y)}{\mathbb{d}\gamma}{\mathbb{d}\delta}}$Where: y is an output of the system; γ is the level to which the outputy has been increased as the input x has been increased; δ is the levelto which the output y has been decreased as the input x has beendecreased; λ(γ, δ) is a density function; and ε_(γδ)(y) is the output ofan imaginary hysteron having parameters γ and δ.
 3. A method accordingto claim 1, wherein determining the required system input furthercomprises determining a value of an input x to the system using thefollowing formula:$x = {\sum\limits_{k = 1}^{n}\left( {x_{\gamma_{k}\delta_{k}} - x_{\gamma_{k}\delta_{k - 1}}} \right)}$where: x_(γ) _(i) _(δ) _(j) is an input of the system that has resultedin the system output y being increased from the minimum output to γ_(i)and then decreased to δ_(j); and δ₀ is the minimum output.
 4. A methodaccording to claim 3, wherein for all required values of x_(γ) _(i) _(δ)_(j) where γ_(i) and δ_(j) are not in the defined plurality of pairs ofvalues, the value of x_(y) _(i) _(δ) _(j) are determined using a processof interpolation using values of x_(γδ) where γ and δ are in the definedplurality of pairs of values.
 5. A method according to claim 4, whereinthe input x is an input voltage of the actuator that deforms thedeformable material, and the output y is a value of a displacement ofthe deformable material.
 6. A method according to claim 1, the methodfurther comprising a process of determining an updated required input tothe system, the determining process comprising: measuring a value of theinput to the system; and determining the updated required input to thesystem using the determined required input to the system and themeasured required input to the system.
 7. A method according to claim 1,the method further comprising a process of determining an updatedrequired input to the system, the determining process comprising:measuring a value of the output of the system to determine a measuredoutput value corresponding to the determined required input; anddetermining the updated required input to the system using the measuredoutput value and the output of the system.
 8. The method of claim 1,wherein the deformable material comprises a deformable mirror.
 9. Acomputer program product comprising a non-transitory computer-readablemedium having instructions encoded thereon and arranged such that whenexecuted by a computer system cause the computer system to perform amethod relating to a system, the system comprising an actuator and oneor more components that exhibit hysteresis including a deformablematerial, the components that exhibit hysteresis having a wiping-outproperty, the system comprising a system input and a system output, thesystem output being dependent on the system input, the methodcomprising: determining an inverse Preisach model for a system, theinverse Preisach model comprising a model input and a model output, themodel output being dependent on the model input, the model inputcorresponding to the system output, the model output corresponding tothe system input; ascertaining a desired system output; determiningusing the desired system output and the inverse Preisach model for thesystem, a required system input corresponding to the desired systemoutput; and applying, to the system, the determined required systeminput to deform the deformable material using the actuator; wherein thestep of determining the inverse Preisach model for the system comprises:defining a plurality of pairs of values, each pair of values comprisinga first value and a second value, each of the first and second valuesbeing less than or equal to a maximum system output and greater than orequal to a minimum system output, the first value being greater than orequal to the second value within each pair; and for each pair of values,determining a system input corresponding to that pair of values, by:increasing the system input, from the minimum system input, until thesystem output is equal to the first value corresponding to therespective pair of values; and thereafter decreasing the system inputuntil the system output is equal to the second value corresponding tothe respective pair of values, the value that the system input isdecreased to being the determined system input corresponding to thatpair of values.
 10. An apparatus, the apparatus comprising: a system,the system comprising an actuator and one or more components thatexhibit hysteresis including a deformable material, the components thatexhibit hysteresis having a wiping-out property, the system comprising asystem input and a system output, the system output being dependent onthe system input; one or more processors configured to: determine aninverse Preisach model for the system, the inverse Preisach modelcomprising a model input and a model output, the model output beingdependent on the model input, the model input corresponding to thesystem output, the model output corresponding to the system input;ascertain a desired system output; determine, using the desired systemoutput and the inverse Preisach model for the system, a required systeminput corresponding to the desired system output; and apply, to thesystem, the determined required system input to deform the deformablematerial using the actuator; wherein determining an inverse Preisachmodel for the system further causes the one or more processors to:define a plurality of pairs of values, each pair of values comprising afirst value and a second value, each of the first and second valuesbeing less than or equal to a maximum system output and greater than orequal to a minimum system output, the first value being greater than orequal to the second value within each pair; and determine, for each pairof values, a system input corresponding to that pair of values, by:increase the system input, from a minimum system input, until the systemoutput is equal to the first value corresponding to the respective pairof values; and thereafter, decrease the system input until the systemoutput is equal to the second value corresponding to the respective pairof values, the value that the system input is decreased to being thedetermined system input corresponding to that pair of values.
 11. Thesystem according to claim 10 comprising an adaptive optics system. 12.The system according to claim 10, wherein the deformable materialcomprises a deformable mirror.